Hostname: page-component-77c89778f8-m8s7h Total loading time: 0 Render date: 2024-07-20T15:40:57.071Z Has data issue: false hasContentIssue false
  • English
  • Français

Particle motions induced by spherical convective elements in Stokes flow

Published online by Cambridge University Press:  21 April 2006

R. W. Griffiths
R. W. Griffiths
Affiliation:
Research School of Earth Sciences, The Australian National University, G.P.O. Box 4, Canberra, A.C.T. 2601, Australia
Get access Rights & Permissions [Opens in a new window]

Abstract

The motions of fluid particles within and around a mass of hot, buoyant material (a thermal) rising through an extremely viscous, unbounded environment are computed using a simple kinematic model. The model is based on a similarity solution by Griffiths (1986a) and allows for growth of thermals due to outward diffusion of heat. Particle motions are also computed for the case of a non-expanding, isothermal sphere, such as a bubble of relatively low-viscosity fluid, in Stokes flow. Motions induced in the surroundings lead to large vertical displacements: the ‘total drift’ function and hydrodynamic mass corresponding to those defined for the inviscid case by Darwin (1953) and Lighthill (1956) are infinite in this unbounded geometry. Rotation of initially horizontal fluid elements (strain) in the surroundings is discussed.

All material lying within an expanding thermal becomes confined at later times to a torus (dye ring) if the Rayleigh number for the thermal is large, to a central tapered blob if Ra < 50, or to an umbrella-shaped cap with narrow stem if Ra takes intermediate values. The ‘mushroom’ shape widely observed for tracers within laminar elements in thermal convection is predicted for intermediate-to-large Rayleigh numbers. Buoyancy and heat, on the other hand, are assumed to remain evenly distributed throughout an enlarging sphere. Laboratory experiments illustrate and confirm the predictions of the model.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 166 , May 1986 , pp. 139 - 159
Copyright
© 1986 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Darwin, C. 1953 Note on hydrodynamics. Proc. Camb. Phil. Soc. 49, 342–354.Google Scholar
Griffiths, R. W. 1986a Thermals in extremely viscous fluids, including the effects of temperature-dependent viscosity. J. Fluid Mech. 166, 115–138.Google Scholar
Griffiths, R. W. 1986b Dynamics of mantle thermals with constant buoyancy or anomalous internal heating. Earth Planet. Sci. Lett. (in press).
Griffiths, R. W. 1986c The differing effects of compositional and thermal buoyancies on the evolution of mantle diapirs. Phys. Earth Planet. Inter. (submitted).
Kojima, M., Hinch, E. J. & Acrivos, A. 1984 The formation and expansion of a toroidal drop moving in a viscous fluid. Phys. Fluids 27, 19–32.Google Scholar
Lamb, H. 1932 Hydrodynamics. Dover.
Lighthill, M. J. 1956 Drift. J. Fluid Mech. 1, 31–53.Google Scholar
Marsh, B. D. 1984 Mechanics and energetics of magma formation and ascension. In Explosive Volcanism: Inception, Evolution and Hazards, pp. 67–83. Washington: National Academy Press.
Marsh, B. D. & Kantha, L. H. 1978 On the heat and mass transfer from an ascending magma. Earth Planet. Sci. Lett. 39, 435–443.Google Scholar
Maxwell, J. C. 1870 On the displacement in a case of fluid motion. Proc. Lond. Math. Soc. 3, 82–87.Google Scholar
Morris, S. 1982 The effects of a strongly temperature dependent viscosity on slow flow past a hot sphere. J. Fluid Mech. 124, 1–26.Google Scholar
Morton, B. R. 1960 Weak thermal vortex rings. J. Fluid Mech. 9, 107–118.Google Scholar
Ramberg, H. 1981 Gravity, Deformation and the Earth's Crust 2nd edn. Academic.
Ribe, N. M. 1983 Diapirism in the Earth's mantle: experiments on the motion of a hot sphere in a fluid with temperature dependent viscosity. J. Volcanol. Geotherm. Res. 16, 221–245.Google Scholar
Schwerdtner, W. M. 1982 Salt rocks as natural analogues of Archaean gneiss diapirs. Geol. Rundschau, 71, 370–379.Google Scholar
Scorer, R. S. 1957 Experiments on convection of isolated masses of buoyant fluid. J. Fluid Mech. 2, 583–594.Google Scholar
Scorer, R. S. 1978 Environmental Aerodynamics. John Wiley.
Sparrow, E. M., Husar, R. B. & Goldstein, R. J. 1970 Observations and other characteristics of thermals. J. Fluid Mech. 41, 793–800.Google Scholar
Turner, J. S. 1957 Buoyant vortex rings. Proc. Roy. Soc. A 239, 61–75.Google Scholar
Turner, J. S. 1964 The flow into an expanding spherical vortex. J. Fluid Mech. 18, 195–208.Google Scholar
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.
Yih, C.-S. 1985 New derivations of Darwin's theorem. J. Fluid Mech. 152, 163–172.Google Scholar